My post, just before yours, answers that question.
Look at it!
Equation 4 is \(\displaystyle 4\frac{\partial}{\partial \eta}\frac{\partial f}{\partial \xi}= 0\).
First, obviously, you can divide both sides by 4 to get
\(\displaystyle \frac{\partial}{\partial \eta}\frac{\partial f}{\partial \xi}= 0\).
Now, to make it easier to see, let \(\displaystyle u= \frac{\partial f}{\partial \xi}\) so that we can write the equation is
\(\displaystyle \frac{\partial u}{\partial \eta}= 0 \) ..................edited
The integral of 0 is, of course, a constant so integrating this equation with respect to \(\displaystyle \eta\) that "constant" might actually be a function of the other variable, \(\displaystyle \xi\). We can write \(\displaystyle u=f(\eta)\) where f can be any differentiable function of \(\displaystyle \eta\).
Since \(\displaystyle u= \frac{\partial f}{\partial \xi}= f(\xi)\). Integrating that \(\displaystyle f= F(\xi)+ G(\eta)\) where F is the integral of f and G is the "constant of integration" which might be a function of the other variable \(\displaystyle \eta\).[/tex]